The Derivative and Limits

Slope and Tangent Lines

The concept of slope is fundamental in calculus, representing the rate of change of a function. For straight lines, slope is constant, but for curves, the slope at a point is given by the derivative.

• Slope-Intercept Form: The equation of a straight line is , where m is the slope and b is the y-intercept.

• Point-Slope Form: , useful for writing the equation of a line given a point and the slope.

• Slope of a Curve: At a specific point, the slope is the derivative of the function at that point.

• Tangent Line: The tangent line to a curve at a point has a slope equal to the derivative at that point.

Example: For , the slope at is .

Limits and the Derivative

Limits are used to define the derivative and analyze the behavior of functions as inputs approach specific values.

• Limit Laws: Fundamental rules for evaluating limits, such as plugging in the value, factoring, and comparing degrees of numerator and denominator.

• Key Steps:

  • First, substitute the value into the function.

  • If indeterminate, factor or simplify.

  • If degrees are the same, compare coefficients.

  • If denominator degree is higher, limit approaches zero.

• Derivative Definition: The slope of the tangent line is the derivative, defined as:

Differentiability and Continuity

Criteria for Continuity

A function is continuous at a point if it meets three criteria:

• 1. exists

• 2. exists

• 3.

Discontinuities can occur due to holes, gaps, or asymptotes.

Differentiability

A function is differentiable at a point if it is continuous there and has no sharp corners or cusps.

• Non-differentiable Points: Occur at cusps, corners, or points of discontinuity.

Rules for Differentiation

Basic Differentiation Rules

Several rules simplify the process of finding derivatives:

• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative.

• Sum/Difference Rule: The derivative of a sum or difference is the sum or difference of the derivatives.

• General Power Rule: For , .

• Chain Rule: Used for composite functions; see below for details.

Product and Quotient Rules

When differentiating products or quotients of functions, special rules apply:

• Product Rule:

• Quotient Rule:

Chain Rule

The chain rule is used to differentiate composite functions, where one function is inside another.

• Chain Rule Formula:

• Application: Identify the "inside function" and differentiate accordingly.

First and Second Derivatives

Interpretation and Applications

The first derivative represents the rate of change (slope), while the second derivative indicates the rate of change of the rate (concavity or acceleration).

• First Derivative: gives the slope or velocity.

• Second Derivative: gives the acceleration or concavity.

• Word Problems: Marginal revenue, cost, and other business applications use derivatives to analyze change.

Example: If is position, then is velocity and is acceleration.

Derivative as Rate of Change

Business Applications

Derivatives are used to approximate changes and analyze rates in business contexts.

• Marginal Revenue: is the approximate change in revenue for a small change in sales.

• Approximating Change:

Continuity and Differentiability Examples

Analyzing Points

To determine if a function is continuous or differentiable at a point:

• Check for continuity: Is the function defined and does the limit exist?

• Check for differentiability: Is there a cusp or corner? If so, the function is not differentiable.

Example: At , a function may be continuous but not differentiable due to a cusp. At , it may not be continuous, hence not differentiable.

Summary Table: Differentiation Rules

The following table summarizes key differentiation rules: